Issue 70

H. Siguerdjidjene et alii, Frattura ed Integrità Strutturale, 70 (2024) 1-23; DOI: 10.3221/IGF-ESIS.70.01

where C is elastoplastic constitutive matrix of FGM. The change in strain is assumed to be divisible into elastic ( e d ε ) and plastic components ( p d ε ) for any increment of stress such that: (17) The strain tensor corresponds to the symmetric part of displacement gradient for an isoparametric element: e p d ε d ε d ε   where B is the strain-displacement matrix of shape function derivatives. The elastic component of increment strain can be written as:   1 e d ε C d σ   (19) where C is the constitutive matrix for three-dimensional linear elastic FGM. For the von Mises yield criteria, the yield surface can be described by:   eq r , σ σ 0 f R     (20) ε Bu sym   u (18)

3 2



 ,



f R  : is the yield function, and eq σ

where

: is the von Mises equivalent stress of FGM, where S is the

S.S

deviatoric part of the Cauchy stress tensor:

1

 

S

ij σ σ δ 3 m

(21)

ij

ij

where m σ is the mean stress, δ ij is the Kronecker delta, and r  is the radius of the yield surface, defined by:

 

   

(22)

R p

r

Y

0

where Y0 σ is the initial yield stress of the FGM, and   R p H .p P  is the isotropic hardening (p is the internal variable corresponding to isotropic hardening and H P is the hardening module). The rule of normality makes it possible to establish the following complementary relations:



f 3 S σ 2 σ

p



  

(23)

d ε

d λ .N

and N

eq

where N: is the gradient of the yield function with respect to the stress tensor, and p d ε : is the plastic strain rate. If the scalar (p), which is the accumulated plastic strain, is defined as the integration of the rate of the accumulated plastic strain during an iterative procedure .

0 t d d    p

 

(24)

p t

where τ is an integration operator, and the rate itself is defined as:

2 3

p p

p d 

(25)

d ε .d ε

9